The region between two concentric spheres of radii a and b… | JEE Main 2016

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Electric Flux & Gauss's Law

The region between two concentric spheres of radii $a$ and $b,$ respectively (see figure), has volume charge density $\rho = \frac{A}{r},$ where $A$ is a constant and $r$ is the distance from the center. At the center of the spheres is a point charge $Q$ . The value of $A$ such that the electric field in the region between the spheres will be constant, is:

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Answer: (c)

Charge within $r$ ( $a < r < b$ ): $q = Q + \int_a^r \frac{A}{r}4\pi r^2 dr = Q + 2\pi A(r^2 - a^2)$ .
$E = \frac{q}{4\pi\varepsilon_0 r^2} = \frac{1}{4\pi\varepsilon_0}\left(2\pi A + \frac{Q - 2\pi A a^2}{r^2}\right)$
For $E$ to be constant the $\frac{1}{r^2}$ term must vanish: $Q = 2\pi A a^2 \Rightarrow A = \frac{Q}{2\pi a^2}$ .
So the answer is (c).

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